Problem 15 - Prealgebra

Question

Solve each of the following problems. What percent of 8 is \(6 ?\)

Step-by-Step Solution

Answer

6 is 75% of 8.

Step 1: Understand the Problem

We need to find out what percentage the number 6 is of the number 8. This involves finding a part-to-whole relationship expressed as a percentage.

Step 2: Write the Fraction

To find out what percent 6 is of 8, we write the situation as a fraction: \( \frac{6}{8} \). Here, 6 is the part and 8 is the whole.

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{6}{8} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. \( \frac{6}{8} = \frac{3}{4} \).

Step 4: Convert to Decimal

Convert the simplified fraction \( \frac{3}{4} \) to a decimal by dividing 3 by 4. The result is 0.75.

Step 5: Convert Decimal to Percentage

To convert the decimal 0.75 to a percentage, multiply by 100. Thus, \( 0.75 \times 100 = 75 \).

Key Concepts

Part-to-Whole Relationship

Understanding the part-to-whole relationship is crucial when dealing with percentages. In our problem, we're focusing on the question: "What percent of 8 is 6?".
We see that number 6 represents the part, while 8 is the whole. Simply put, we're trying to find out what piece or percentage the number 6 occupies out of the number 8.
This concept is foundational in percentage calculations, allowing us to frame the problem initially as a fraction. Once understood, finding the percentage becomes a straightforward task.

Fraction Simplification

Fraction simplification is a vital step in making calculations easier and more manageable. When we originally write the problem as a fraction \( \frac{6}{8} \), we are acknowledging 6 as the part and 8 as the whole.
However, working with simpler terms is always ideal. Thus, simplifying \( \frac{6}{8} \) involves dividing both the numerator and the denominator by their greatest common divisor, which in this case is 2. As a result, \( \frac{6}{8} \) is simplified to \( \frac{3}{4} \).
This simplification makes future steps in the problem more straightforward and less prone to error.

Decimal Conversion

Converting a simplified fraction into a decimal expands its utility and is vital for percentage calculations. Once we have simplified the fraction to \( \frac{3}{4} \), we proceed to a decimal conversion.
This is achieved by dividing the numerator by the denominator. Hence, \( 3 \div 4 = 0.75 \).
Decimals are integral in various mathematical contexts, providing an alternative to represent proportions and facilitating the transition to percentage form.

Percentage Calculation

After converting our fraction to a decimal, the final step is to calculate the percentage. This involves a straightforward operation.
We simply multiply the decimal 0.75 by 100 to find out the percentage. Therefore, \( 0.75 \times 100 = 75 \).
This calculation reveals that 6 is 75% of 8, completing our initial objective and illustrating the power of understanding relationships between numbers.